2025/12/24 by Yiqi Huang, Tristan Ozuch, Huang, Yiqi +1
Mathematics · #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Geometry and complex manifolds
paper · doi:10.48550/arxiv.2512.21317
We refine the regularity of noncollapsed limits of 5-dimensional manifolds with bounded Ricci curvature. In particular, for noncollapsed limits of Einstein 5-manifolds, we prove that (1) tangent cones are unique of the form ℝ×ℝ4/Γ on the top stratum, hence outside a countable set of points; this follows from a new isolation result for cones of the form ℝ×ℝ4/Γ among all tangent cones, (2) the singular set is entirely contained in a countable union of Lipschitz curves and points, (3) away from a nowhere dense subset, these Lipschitz curves consist of smooth geodesics, (4) the interior of any geodesic is removable: limits of Einstein manifolds are real-analytic orbifolds with singularities along geodesic and bounded curvature away from their extreme points, and (5) if an asymptotically Ricci-flat 5-manifold with Euclidean volume growth has one tangent cone at infinity that splits off a line, then it is the unique tangent cone at infinity. These results prompt the question of the orbifold regularity of noncollapsed limits of Einstein manifolds off a codimension 5 set in arbitrary dimension. The proofs rely on a new result of independent interest: all spherical and hyperbolic 4-orbifolds are isolated among Einstein 4-orbifolds in the Gromov-Hausdorff sense. This yields various gap theorems for Einstein 4-orbifolds, which do not extend to higher dimensions. The proofs of these gap theorems require a careful analysis of singular metrics and families of metrics that degenerate.